I feel that we may be ignoring the psychological aspects of having a strong BATNA here. It's easy to say that you should just "ask for a better offer", but it is waay easier to actually do that when you know how much you are worth to another employer.

Some people may be able to pull off something like what you describe, but all research I've read on negotiating indicate that knowing your BATNA and having a strong one is key to achieving good negotiation results.